1078 
MR. r. W. DTSOX ON THE POTENTIAL OF AN ANCHOR RING. 
^ K 1 _o ^ 
7—T = -R — COS 2o — 
4:71 ah 2 ^ 16c^ 
, /K — 1 1 „ ^ \ o cosli u cos V 
+ f —^7— — \.e~ cos 2v 
2c 
+ 
— (e cos Sp — 3e " cos v) + — 
8c 
' 1 a cosh u cos v 
Jc 2c2 
a 
~ ^2 (K - cos 4v) 
. a? /Iv , „ \ 5 cosh ho cos^ v — sinh hi sin^ v 
+ ^ 2 - -16- 
— (11 cosh® w cos® V — 5 sinh® u sin® v) 
ct 
-f (e“®“ cos 3v — 3e"“ cos v) 
10 cosh u cos V llft^ 
12c2 
16 
128c2 
+ ^(K»ie-''cos Av)il, 
Therefore 
Yj2TTab = K — e ®^' cos 2v 
H— •{ cos V 
C 
^ cosh u — 
4c2“ + 9 + c 
16 
3e“ — e““ 
48 
e ®" cos 3v 
I if Jt 7 4 cosh2zf + 7 _ 3 _ o?;_ 339 _ ^-•Zu _ 3 _ „-i!/ 
cM 64 326 .256 e 
} 
a 
+ ~ COS 2v -iK 
3 cosh 2 m + 2 
32 
I 1 p'Zu I iA I 10 7 ^ —3« I _1_ fi—iu 
^ 2 ^ ^ 32 ^ 192 ^ ^ 96 ^ 
f6“ 
c- 
+ - COS 4^; <1 — ^6 - e- 
2w 
_L 
96 ^ f • 
. . . (51). 
At the surface of the ring 
16c 
u = {), K = log —, 
and 
a- 
V = 2^a6 ^Kll + ii~) - fif ^ + cosv(- - I'l- 
( 12 . ' ^ “ y 2 ® / c 
+ cos 21! 
~ i + 
^ I 27i\ a' 
32 1 9 2 Ks 
a 
— — COS 3 y 
24c 
a- 
^ cos 4ii 
(52). 
§ 28. Let us take Saturn to be a sphere of radius r and of density p. Assume 
that the ring is fluid rotating with angular velocity (o, of density <x; let its section 
be supposed elliptic (semi-axes a and h), h being much smaller than a : and let its 
mean radius be c. 
